In this paper, we prove the following non-linear generalization of the
classical Sylvester-Gallai theorem. Let K be an algebraically closed
field of characteristic 0, and F={F1β,β―,Fmβ}βK[x1β,β―,xNβ] be a set of irreducible homogeneous polynomials of
degree at most d such that Fiβ is not a scalar multiple of Fjβ for iξ =j. Suppose that for any two distinct Fiβ,FjββF, there is kξ =i,j such that Fkββrad(Fiβ,Fjβ). We prove that such radical SG
configurations must be low dimensional. More precisely, we show that there
exists a function Ξ»:NβN, independent of
K,N and m, such that any such configuration F must
satisfy
dim(spanKβF)β€Ξ»(d).
Our result confirms a conjecture of Gupta [Gup14, Conjecture 2] and
generalizes the quadratic and cubic Sylvester-Gallai theorems of [S20,OS22].
Our result takes us one step closer towards the first deterministic polynomial
time algorithm for the Polynomial Identity Testing (PIT) problem for depth-4
circuits of bounded top and bottom fanins. Our result, when combined with the
Stillman uniformity type results of [AH20a,DLL19,ESS21], yields uniform bounds
for several algebraic invariants such as projective dimension, Betti numbers
and Castelnuovo-Mumford regularity of ideals generated by radical SG
configurations.Comment: 62 pages. Comments are welcome